Classical linear algebra is done over a real-closed field or an algebraically closed field of characteristic ; the latter is the simplest context, which is now pretty thoroughly understood. Traditionally, we take these fields to be the field of real numbers and the field of complex numbers, although arguably we only really use algebraic numbers. (From a constructive point of view, some of the classical material is valid only over discrete fields, so we must restrict attention to algebraic numbers, or to some discrete extension, for these results to hold in their classical form.)
Fancier linear algebra is done over incomplete fields and fields with positive characteristic (and constructively over nondiscrete fields). Sometimes a generalization to categories of finitely generated projectives over a ring is considered. In infinite dimensions one rarely studies purely algebraic version, which is considered as a linear algebra, but more often one equips them with topological structure, what enters the subject of functional analysis.
If one is interested in tensor products as well, then one gets a generalization called multilinear algebra: tensor algebra, tensors, exterior and symmetric algebras are some of the main characters in that theory. Study of determinants is important in the usual linear algebra but it is also closely related to the study of exterior algebras.